Handbook on Plasma Instabilities
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They measured the final state of the beam-plasma instability, and found good agreement with QL theory for the final spectrum and velocity distribution. A few years later, the nonlinear evolution of the electron beam-plasma instability started to be investigated . The most recent experimental results on strong Langmuir turbulence in the beam plasma interaction have been presented by Wong and collaborators .
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They explore the time evolution of the electrostatic wave spectra, during several ion plasma periods. In this paper we determine experimental conditions for observing strong Langmuir turbulence in our device.
We obtain the time growth rates for the excited waves and compare them with existing theory. In this regime, we detect the presence of electron wave with frequency larger than the beam resonant wave. We show that for this higher frequency wave Landau damping acts effectively. The paper is organized as follows. In Section 2 we describe the experimental apparatus and the diagnostics. In Section 3 we present the experimental conditions for the excitement of non-linear phenomena. We report and examine the excited electron wave in Section 4.
Section V presents our conclusions. A quiescent plasma is generated independently in both chambers by accelerating primary electrons produced by tungsten hot cathode filaments. An electron beam is created biasing the source chamber negatively with respect to the grounded target chamber. The electron density and temperature measurements were performed using a single cylindrical Langmuir probe. The temperature was determined by the traditional method. The density was obtained by detecting the cut-off frequency of an electromagnetic wave launched into the plasma.
The typical plasma potential in the target plasma is positive and of the order of V. For Langmuir electron wave detection a thin wire working as a dipole probe, insulated by a capacitor, is connected to the spectrum analyzer. The evolution of the electron energy distribution function was measured by an electrostatic multigrid energy analyzer using the retarding potential method .
Table 1 shows typical plasma parameters for our experiment. At the range of working pressure the collisional damping rate due to electron-ion and electron-neutral collisions is smaller than the generation rate of waves, the beam-plasma instability growth rate, g bp. This leads to the condition.
Once this condition is satisfied, the excitation of Langmuir wave by the beam-plasma instability is allowed. Using the parameters shown in Table 1 we see that Equation 1 is satisfied.
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The beam tends to disturb the plasma, so we consider measurements for distances z larger than 0. At this distance the plasma is uniform in density and temperature, as shown in Fig. The waves are still far from saturation when they enter the uniform region, and all important physics occurs in that region. In order to explain the observed waves in our experiment, we need a comprehensive theory. Such theory requires the specification of a free-energy source, a mechanism for wave growth driven by this free-energy source, and a wave energy saturation mechanism.
Free energy for the Langmuir wave growth is provided by the beam relaxation. For the range of parameters used in the experiment the Langmuir wave growth is driven primarily by the kinetic version of the beam instability. From our experimental results it is possible to establish a threshold line for the transition between QL and SLT regimes. According to quasi-linear theory, and considering pile up effects, the maximum normalized energy density of the waves is given by. Stabilization of maximum wave amplitude suggests the appearance of a nonlinear phenomenon that interferes with the beam-plasma instability evolution.
Together with the saturation of wave energy we see the appearance of ion-sound waves. From Fig.
These points are shown in Fig. The line shown in Fig. For a system with parameters in the region below the MI excitation threshold, the Langmuir wave energy saturates due to the complete relaxation of the beam. The position of saturation of the wave energy corresponds to the position of plateau formation in the distribution function, as predicted by quasi-linear theory known as ''beam relaxation length''. For a system with parameters in the region above the MI excitement threshold, the nonlinear instability transports the plasma waves to shorter wavelength scales higher frequencies where the resonant interaction between beam and waves does not occur.
As a consequence the beam propagates for larger distances than the ones predicted by QL theory. The axial evolution of the electron beam distribution function shows clearly the different behaviour for the quasi-linear and for the nonlinear regime, as we can observe in Fig. The length l is the distance between the grid and the axial position where the plateau is formed in the distribution function.
The lines shown in Fig. To illustrate the evolution of the excited Langmuir waves across the system we show the spectra of the electron waves in Fig. Plots of the spectrum were obtained in three different axial positions. Harmonics of the main frequency have already been observed. The second harmonic is still present with smaller amplitude than in case B. The wave amplitude increases with the distance from the grid and reaches a maximum value at z 0. The space evolution of the Langmuir wave amplitude determines the behaviour of the system.
In this section we concentrate on the electron waves observed in the nonlinear regime. The results obtained from the experimental date are compared with the existing theory. It is important to point out that we have a continuous beam injection. This fact leads us to a situation where close to the grid, at the point of beam injection, we expect that the excited wave amplitude grows as predicted by QL theory. After reaching a certain threshold, the excited wave can became a pump wave for wave-wave interactions.
Starting from the space profile along z of the wave amplitude we obtain the spatial growth rate, k , for the instability and from that we obtain the temporal growth rate, g. Notice that with these parameters we are above the threshold MI line shown in Fig. Observe that the higher frequency wave only appears after the Langmuir wave has reached region labeled by 2. Notice that from our experimental results we can only obtain the spatial growth rate. Ihr PDF motion crystallography place.
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